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Chapter 2: Problem 7

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$

Short Answer

Expert verified
The asymptotes of the given function \(r(x) = \frac{9x + 5}{x^2 - x - 6}\) are vertical asymptotes at \(x = 3\) and \(x = -2\) and a horizontal asymptote at \(y = 0\).

Step by step solution

01

Factor the denominator

In order to find the vertical asymptotes, we need to factor the denominator of the given function: \[ r(x) = \frac{9x + 5}{x^2 - x - 6} \] Now, we'll factor the denominator: \[ x^2 - x - 6 = (x - 3)(x + 2) \]
02

Determine the vertical asymptotes

We can now find the vertical asymptotes by setting the denominator equal to zero and solving for the variable x: \[ (x - 3)(x + 2) = 0 \] This means that \(x = 3\) and \(x = -2\) are the vertical asymptotes.
03

Compare degrees of the numerator and denominator

Next, we need to find the horizontal asymptotes by comparing the degrees (highest powers) of the numerator and denominator of the given function: Degree of numerator: 1 (9x) Degree of denominator: 2 (x^2) Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote will be \(y = 0\) (the x-axis). Thus, the asymptotes of the given function are as follows: Vertical asymptotes: \(x = 3\) and \(x = -2\) Horizontal asymptote: \(y = 0\)

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